標本分散の期待値

$$\begin{array}{rcl} s^2&=&\displaystyle \frac{1}{n}\sum_{k=1}^{n}\left( \displaystyle X_k - \overline{X} \displaystyle \right)^2\,\dotso\,標本分散(sample \, variance)\\ E\left[s^2\right]&=&E\left[ \displaystyle\frac{1}{n}\sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]\\ &=&\displaystyle \frac{1}{n} E\left[ \sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]\\ &=&\displaystyle \frac{1}{n} \left(n-1\right)\sigma^2 \,\dotso\,\displaystyle \href{https://shikitenkai.blogspot.com/2019/07/overlinex2.html}{E\left[\sum_{k=1}^{n} \left(X_k -\overline{X}\right)^2 \right]=\left(n-1\right)\sigma^2}\\ &=&\displaystyle \frac{n-1}{n}\sigma^2 \\ \end{array}$$